As per the equation, 2 raised to the power of x equals 1.5. We're interested in finding a range in which x could possibly lie.
In order to ascertain this, what we can do is to apply a property of logarithms to this equation. The property says: if we have an equation b^(c) = a, it can also be represented as log_b(a) = c.
Applying this property to our equation 2^(x) = 1.5, it becomes log_2(1.5) = x.
Now, we will determine the interval in which the solution x could possibly reside. We know the value of log_2(1) and log_2(2). The first one equals 0 and the second one equals 1. Given that 1.5 falls between the numbers 1 and 2, we deduce that the resultant x would be a number between 0 and 1. Therefore, the lower limit of the interval is 0 and the upper limit is 1.
After performing the calculation using logarithms, we find that the precise value of x is approximately 0.5849625007211562.
In conclusion, the solution x to the equation 2^(x)=1.5 falls within the interval [0, 1], with an approximate value of x = 0.5849625007211562.